In the 1995 women’s figure-skating world championship, Chen Lu, Nicole Bobek, and Surya Bonaly were in first, second, and third place after they finished skating. Then 14-year-old Michelle Kwan surged into fourth with a strong showing in the free skate. In a bizarre twist attributed to an unusual new judging system, Kwan’s strong performance caused Bonaly and Bobek to switch places. Bonaly got the silver, Bobek got the bronze, and Nobel Prize-winning economist Kenneth Arrow got even more vindication for his work on the drawbacks of rank-order voting.

In the 1950s, Arrow had argued that there is no “good” election method. His research focused on methods in which each voter lists some or all of the candidates in a strict linear order (first choice, second choice, etc.). Arrow proved that any such method must display at least one of the following unreasonable characteristics: (1) a “dictator” always decides who wins, regardless of how everybody else votes; (2) outcomes sometimes conflict with even unanimous electoral preferences; or (3) a seemingly irrelevant candidate changes the relative standing of two others.

The 1995 championship illustrated the third problem. Previously, judges scored skaters from 0 to 6, and the one with the highest average won. But that year, scores were used only to produce rank orders. Final standings were determined by a complex algorithm based largely on the number of times each skater was ranked first, second, third, and so forth. When some judges gave Kwan a higher score than Bobek, some of Bobek’s rankings slipped. So even though her numeric scores didn’t change, she was demoted to bronze.

Arrow’s theorem could have predicted this oddity. But when he concluded that all election methods are inherently flawed, he had neglected an important fact: election methods do not have to be based on rank ordering.

Honeybees hold “elections” each year to choose a new location for their hive; bad decisions could lead to the colony’s annihilation. Over 50 million years, natural selection produced a system in which scout bees “score” each candidate site with dances describing the site’s direction and distance. The more intense the dance, the greater the chance that other scouts will investigate the site. When a site attracts a sufficiently large majority of followers, it wins.

Sparta, the longest-lasting substantially democratic government in history, voted in a similar way from about 700 b.c.e. until at least 220 b.c.e. Spartans elected Gerontes and Ephors (council members who had the power to dethrone kings) by means of a shouting system. The candidate with the loudest support won.

Both the bees’ system and the Spartans’ are examples of range voting: each voter scores each candidate within a given range (say, 0 to 99); the one with the highest total wins. As John Harsanyi (also a Nobelist) observed when Arrow’s research was published, range voting accomplishes what Arrow deemed impossible. But since his point went against 1950s economic gospel, it was ignored.